Answer Equivalence in Calculus Courseware

We know that oftentimes in calculus, there’s more than one way to solve a problem. While some online systems don’t allow for multiple correct answers, Hawkes Learning’s courseware was built by subject matter experts who painstakingly went through examples to ensure students are given credit for equivalent answers.

Marvin, one of our lead calculus content editors, explained why it’s so important to include equivalent answers in the courseware: “There are often different methods of solving, and we don’t want to penalize students for getting a correct answer. When that happens, students get frustrated and doubt themselves. We want to boost their confidence.”

Sample Problem from Trigonometric Integrals

The first two correct answers are generated using Method 1 of solving, while the next three are generated using Method 2 of solving.

Problem

Evaluate the indefinite integral ∫ 7tan(4x)sec6(4x)dx. Use C for the constant of integration. Write the exact answer. Do not round.

Correct Answer 1

Method 1: We can use u-substitution with u = sec(4x) after rewriting the integral as
7 ∫ sec5(4x) · sec(4x)tan(4x)dx. Note that the answer has the fraction 7/24 as the coefficient of the secant function.

Correct Answer 2

Method 1: We can use u-substitution with u = sec(4x) after rewriting the integral as
7 ∫ sec5(4x) · sec(4x) tan(4x)dx. Note that the answer has the secant function as part of the numerator of the answer.

Correct Answer 3

Method 2: We can use u-substitution with u = tan(4x) after rewriting the integral as
7 ∫ tan(4x) [1 + tan2(4x)]2sec2(4x)dx. Note that the answer has several terms with tangent and fractional coefficients.

Correct Answer 4

Method 2: We can use u-substitution with u = tan(4x) after rewriting the integral as
7 ∫ tan(4x) [1 + tan2(4x)]2sec2(4x)dx. Note that the answer has the fraction 7/8 factored out.

Correct Answer 5

Method 2: We can use u-substitution with u = tan(4x) after rewriting the integral as
7 ∫ tan(4x) [1 + tan2(4x)]2sec2(4x)dx. Note that the answer has the fraction 7tan2(4x)/8 factored out.

Correct Answers 6 & 7

If students rewrite the integrand in terms of sine and cosine and work it out correctly, credit is also given.

Below are two examples of a student answering the problem using cos(4x).

Sample Problem from The Chain Rule

This question shows the application of the Chain Rule, and the correct answer can be written in different ways as shown below.

Problem

Find the derivative of the function F(x) = – 3(13 + 2√x)-5.

Correct Answer 1

The student applies the Chain Rule and writes the last factor as 1/√x.

Correct Answer 2

The student applies the Chain Rule and writes the last factor as x-1/2.

Correct Answer 3

The student applies the Chain Rule and rewrites the square root of x in terms of fractional exponents.

Correct Answer 4

The student applies the Chain Rule and rewrites the whole answer as one fraction using the positive exponent 6 for the expression in parentheses.

Correct Answer 5

The student applies the Chain Rule and rewrites the answer as one fraction using the exponent of negative 6 for the expression in parentheses.

Sample Problem from Integration by Parts

Problem

Evaluate the integral ∫(t + 1)e4tdt. Use C for the constant of integration. Write the exact answer. Do not round. (Hint: Use an alternative method if integration by parts is not required.)

2 thoughts on “Answer Equivalence in Calculus Courseware”

For the first example involving trigonometric integrals, you did cover many of the possible answers. However, a common approach by students is to rewrite the integrand in terms of sine and cosine. In that case, let u = cos(4x) and the result would be (7/24)(cos(4x))^-6 + C.

Great point, Professor Belloit! Our courseware does accept that as correct. I’ve updated the blog post with a few screenshots addressing that method. I appreciate you looking at the answers so closely! Thank you, and let us know if you have more questions.